Ramification des séries formelles

2004 ◽  
Vol 47 (2) ◽  
pp. 237-245
Author(s):  
François Laubie
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Characteristic P ◽  
Substitution Law

AbstractLet p be a prime number. Let k be a finite field of characteristic p. The subset X + X2k[[X]] of the ring k[[X]] is a group under the substitution law ○ sometimes called the Nottingham group of k, it is denoted by Rk. The ramification of one series γ ∈ Rk is caracterized by its lower ramification numbers: , as well as its upper ramification numbers:By Sen's theorem, the um(γ) are integers. In this paper, we determine the sequences of integers (um) for which there exists γ ∈ Rk such that um(γ) = um for all integer m & 0.

scholarly journals Bounded realization of l-groups over global fields

1998 ◽  
Vol 150 ◽  
pp. 13-62 ◽  
Author(s):  
Wulf-Dieter Geyer ◽  
Moshe Jarden
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Finite Fields ◽  
Galois Extension ◽  
Prime Ideals ◽  
Transfer Principle ◽  
Global Fields ◽  
Regular Extension ◽  
Root Of Unity ◽  
Result Theorem

Abstract.We use the method of Scholz and Reichardt and a transfer principle from finite fields to pseudo finite fields in order to prove the following result. THEOREM Let G be a group of order ln, where l is a prime number. Let K0be either a finite field with |K0| > l4n+4or a pseudo finite field. Suppose that l ≠ char(K0) and that K0does not contain the root of unity ζl of order l. Let K = K0(t), with t transcendental over K0. Then K has a Galois extension L with the following properties: (a) (L/K) ≅ G; (b) L/K0is a regular extension; (c) genus(L) < ; (d) K0[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K0: K0]; (e) (t)∞totally decomposes in L; (f) L = K(x), withand deg(ai(t)) < deg(a1(t)) for i = 1,…,ln.

scholarly journals Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs

1980 ◽  
Vol 32 (3) ◽  
pp. 714-733 ◽  
Author(s):  
N. B. Tinberg
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Modular Representations ◽  
Characteristic P ◽  
Representations Of Finite Groups ◽  
A Finite Group

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”(ii) H= ⋂n∈N n-1Bn(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

scholarly journals A search for c-Wieferich primes

2020 ◽  
pp. 1-18
Author(s):  
Alex Samuel Bamunoba ◽  
Jonas Bergström
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Number Formula ◽  
Existence Criterion

Let [Formula: see text] be a power of a prime number [Formula: see text], [Formula: see text] be a finite field with [Formula: see text] elements and [Formula: see text] be a subgroup of [Formula: see text] of order [Formula: see text]. We give an existence criterion and an algorithm for computing maximally [Formula: see text]-fixed c-Wieferich primes in [Formula: see text]. Using the criterion, we study how c-Wieferich primes behave in [Formula: see text] extensions.

scholarly journals Cryptography based on the Matrices

2017 ◽  
Vol 37 (3) ◽  
pp. 75-83 ◽  
Author(s):  
M. Zeriouh ◽  
A. Chillali ◽  
Abdelkarim Boua
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
New Method ◽  
Zero Matrix ◽  
The Matrix ◽  
First Time ◽  
Cryptographic Scheme

In this work we introduce a new method of cryptography based on the matrices over a finite field $\mathbb{F}_{q}$, were $q$ is a power of a prime number $p$. The first time we construct thematrix $M=\left(\begin{array}{cc}A_{1} & A_{2} \\0 & A_{3} \\\end{array}\right)$ were \ $A_{i}$ \ with $i \in \{1, 2, 3 \}$ is the matrix oforder $n$ \ in \ $\mathcal{M}(\mathbb{F}_{q})$ - the set ofmatrices with coefficients in $\mathbb{F}_{q}$ - and $0$ is the zero matrix of order $n$. We prove that $M^{l}=\left(\begin{array}{cc}A_{1}^{l} & (A_{2})_{l} \\0 & A_{3}^{l} \\\end{array}\right)$ were $(A_{2})_{l}=\sum\limits_{k=0}^{l-1}A_{1}^{l-1-k}A_{2}A_{3}^{k}$ for all $l\in \mathbb{N}^{\ast}$. After we will make a cryptographic scheme between the two traditional entities Alice and Bob.

Characteristic p Galois Representations That Arise from Drinfeld Modules

2000 ◽  
Vol 43 (3) ◽  
pp. 282-293 ◽  
Author(s):  
Nigel Boston ◽  
David T. Ose
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Galois Representations ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Absolute Galois Group ◽  
Characteristic P ◽  
Finite Characteristic ◽  
The Absolute

AbstractWe examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group’s action on the division points of an appropriate Drinfeld module.

Fermat Jacobians of Prime Degree over Finite Fields

1999 ◽  
Vol 42 (1) ◽  
pp. 78-86 ◽  
Author(s):  
Josep González
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Cyclotomic Field ◽  
Roots Of Unity ◽  
Characteristic P ◽  
Prime Degree ◽  
Numerical Criterion

AbstractWe study the splitting of Fermat Jacobians of prime degree l over an algebraic closure of a finite field of characteristic p not equal to l. We prove that their decomposition is determined by the residue degree of p in the cyclotomic field of the l-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

scholarly journals Right-angled Artin groups and enhanced Koszul properties

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alberto Cassella ◽  
Claudio Quadrelli
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Galois Cohomology ◽  
Finite Graph ◽  
Galois Groups ◽  
Cohomology Algebra ◽  
Artin Group ◽  
Artin Groups ◽  
Koszul Algebra ◽  
Right Angled Artin Groups

AbstractLet 𝔽 be a finite field. We prove that the cohomology algebra H^{\bullet}(G_{\Gamma},\mathbb{F}) with coefficients in 𝔽 of a right-angled Artin group G_{\Gamma} is a strongly Koszul algebra for every finite graph Γ. Moreover, H^{\bullet}(G_{\Gamma},\mathbb{F}) is a universally Koszul algebra if, and only if, the graph Γ associated to the group G_{\Gamma} has the diagonal property. From this, we obtain several new examples of pro-𝑝 groups, for a prime number 𝑝, whose continuous cochain cohomology algebra with coefficients in the field of 𝑝 elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-𝑝 Galois groups of fields formulated by J. Mináč et al.

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